We investigate the concept of a standard map for the interaction of relativistic particles and electrostatic waves of arbitrary amplitudes, under the action of external magnetic fields. The map is adequate for physical settings where waves and particles interact impulsively, and allows for a series of analytical result to be exactly obtained. Unlike the traditional form of the standard map, the present map is nonlinear in the wave amplitude and displays a series of peculiar properties. Among these properties we discuss the relation involving fixed points of the maps and accelerator regimes.

We analyze the behavior of a relativistic particle moving under the influence of a uniform magnetic field and a stationary electrostatic wave. We work with a set of pulsed waves that allows us to obtain an exact map for the system. We also use a method of control for near-integrable Hamiltonians that consists of the addition of a small and simple control term to the system. This control term creates invariant tori in phase space that prevent chaos from spreading to large regions, making the controlled dynamics more regular. We show numerically that the control term just slightly modifies the system but is able to drastically reduce chaos with a low additional cost of energy. Moreover, we discuss how the control of chaos and the consequent recovery of regular trajectories in phase space are useful to improve regular particle acceleration.

We analyze the dynamics of a relativistic particle moving in a uniform magnetic field and perturbed by a standing electrostatic wave. We show that a pulsed wave produces an infinite number of perturbative terms with the same winding number, which may generate islands in the same region of phase space. As a consequence, the number of isochronous island chains varies as a function of the wave parameters. We observe that in all the resonances, the number of chains is related to the amplitude of the various resonant terms. We determine analytically the position of the periodic points and the number of island chains as a function of the wave number and wave period. Such information is very important when one is concerned with regular particle acceleration, since it is necessary to adjust the initial conditions of the particle to obtain the maximum acceleration. PACS number(s): 05.45.-a, 52.20.Dq, 45.50.DdWave-particle interaction is basically a nonlinear process [1,2] that may present regular and chaotic trajectories in its phase space [3]. This kind of interaction can be found in many areas of physics [1,[4][5][6], and it is used in a wide range of applications as an efficient way for particle heating [1, 7-9] and particle acceleration [1,5,[9][10][11].References [12][13][14] present two cases of particles moving in a uniform magnetic field and perturbed by electrostatic waves. For such systems, appropriate resonant conditions are responsible for a great amount of particle acceleration. References [12,13] determine the parameters values for which the acceleration may be maximum.However, the process of regular acceleration also depends on the trajectory followed by the particles. To attain the condition of maximum acceleration, it is necessary to know the position of the resonances and the number of island chains as a function of the parameters. In this way, it is possible to properly adjust the initial conditions of the particles and make them follow the best trajectory in phase space. Moreover, successive bifurcations changing the number of chains modify the acceleration conditions. Even so, these bifurcations have not been explored yet.In this report, we analyze the onset of different island chains described by the dynamics of a twist system consisting of a relativistic particle moving in a uniform magnetic field. This integrable system is kicked by standing electrostatic pulses [15][16][17], such that it becomes nearintegrable for small amplitudes [1-3, 12, 18].Expanding the pulses in a Fourier-Bessel series, we observe that the wave presents an infinite number of perturbative terms. There are groups of perturbative terms with the same winding number that may generate different islands in the same region of phase space. This superposition alters the number of island chains according to the wave parameters. Using the map of the system, we carry a series of analytical estimates, including the position of the periodic points in phase space and the number of island chains as functions of the wave parameters.

Intrinsically nonlinear coupled systems present different oscillating components that exchange energy among themselves. We present a new approach to deal with such energy exchanges and to investigate how it depends on the system control parameters. The method consists in writing the total energy of the system, and properly identifying the energy terms for each component and, especially, their coupling. To illustrate the proposed approach, we work with the bi-dimensional spring pendulum, which is a paradigm to study nonlinear coupled systems, and is used as a model for several systems. For the spring pendulum, we identify three energy components, resembling the spring and pendulum like motions, and the coupling between them. With these analytical expressions, we analyze the energy exchange for individual trajectories, and we also obtain global characteristics of the spring pendulum energy distribution by calculating spatial and time average energy components for a great number of trajectories (periodic, quasi-periodic and chaotic) throughout the phase space. Considering an energy term due to the nonlinear coupling, we identify regions in the parameter space that correspond to strong and weak coupling. The presented procedure can be applied to nonlinear coupled systems to reveal how the coupling mediates internal energy exchanges, and how the energy distribution varies according to the system parameters.

We investigate the interaction of electromagnetic waves and electron beams in a 4 m long traveling wave tube (TWT). The device is specifically designed to simulate beam-plasma experiments without appreciable noise. This TWT presents an upgraded slow wave structure (SWS) that results in more precise measurements and makes new experiments possible. We introduce a theoretical model describing wave propagation through the SWS and validated by the experimental dispersion relation, impedance, and phase and group velocities. We analyze nonlinear effects arising from the beam–wave interaction, such as the modulation of the electron beam and the wave growth and saturation process. When the beam current is low, the wave growth coefficient and saturation amplitude follow the linear theory predictions. However, for high values of current, nonlinear space charge effects become important and these parameters deviate from the linear predictions, tending to a constant value. After saturation, we also observe trapping of the beam electrons, which alters the wave amplitude along the TWT.